The tilting property for $F_{\ast }^e{\mathcal{O}}_X$ on Fano surfaces and threefolds

Let $X$ be a smooth variety over a field of characteristic $p$. It is a natural question whether the Frobenius pushforwards $F_{\ast }^e{\mathcal{O}}_X$ of the structure sheaf are tilting bundles. We show if $X$ is a smooth del Pezzo surface of degree $\le 3$ or a Fano threefold with $\mathrm{v}\mathrm{o}\mathrm{l}(K_X)<24$ over a field of characteristic $p$, then ${\mathrm{E}\mathrm{x}\mathrm{t}}^i(F_{\ast }^e{\mathcal{O}}_X,F_{\ast }^e{\mathcal{O}}_X)\mathrm{\ne }0$ and thus $F_{\ast }^e{\mathcal{O}}_X$ is not tilting.